Set Operations - SS1 Mathematics Lesson Note

Union of sets

The union of two sets A and B is the set that contains all the elements of A and B, A∪B = {x|x∈A or x∈B}.

If A={2,3,4} and B={5,6,7}, then the union of A and B →A∪B={2,3,4,5,6,7}.

Intersection of sets

The intersection of two sets \(X\) and \(Y\) is the set \(Z\) containing all elements that are common to both \(X\) and \(Y\), that is \(X \cap Y\ = \ \{ a:\ a \in X\ and\ a \in Y\}\).

If \(X = \{ 5,11,9\}\) and \(Y = \{ 5,6,7\}\), then the intersection of \(A\) and \(B\ \rightarrow A \cap B = \{ 5\}\).

Disjoint sets

When two sets \(K\) and \(L\) have no members in common, they are said to be disjoint sets and their intersection is empty or null.

If \(X = \{ 13,23,33\}\) and \(Y = \{ 55,66,77\}\), then the intersection of \(X\) and \(Y\ \rightarrow X \cap Y = \{\}\), meaning \(X\) and \(Y\) are disjoint sets.

Difference of sets

The difference of two sets \(A\) and \(B\) denoted as \(A - B\) is the set of all elements contained in \(A\) which are not in \(B\). That is, \(A - B = \{ x:\ x \in A\ and\ x \notin B\}\). Likewise, \(B - A = \{ x:\ x \in B\ and\ x \notin A\}\). Note, \(B - A \neq A - B\).

If \(A = \{ 5,3,7\}\) and \(B = \{ 5,6,7\}\), then \(B - A = \left\{ x:\ x \in B\ and\ x \notin A \right\} = \{ 6\}\) and \(A - B = \left\{ x:\ x \in A\ and\ x \notin B \right\} = \{ 3\}\). It is clear that \(B - A \neq A - B\).

Complements of sets

Consider the universal set \(U\) and a set \(A\), the universal complement of \(A\) or \(A\) complement is the set of all elements not contained in \(A\) but part of the universal set \(U\). Here, \(A^{'}\) or \(A^{c} = \{ x:x \in U\ and\ x \notin A\}\).

If \(U = \left\{ 1,2,3,4,5,6,7 \right\}\ and\ A = \{ 5,6,7\}\), then \(A^{'} = \{ 1,2,3,4\}\).